Rigidity conjecture for circle maps of an oval

Let $\gamma$ be an oval. For directions $u$ and $v$, let $f_u,f_v:\gamma\to\gamma$ be the involutions induced by the family of parallel lines in those directions, and set $F_{u,v}=f_u\circ f_v$. Rigidity conjecture. Suppose that $F_{u,v}:\gamma\to\gamma$ is conjugated to a rotation for all pairs of directions $u,v$. Then $\gamma$ is an ellipse. This conjecture concerns the characterization of ellipses among ovals by the rotation-conjugacy of their associated circle maps; it is open as far as the source indicates.